Internat. J. Math. & Math. Sci.
Vol. 22, No. 3 (1999) 597–604
S 0161-17129922597-5
© Electronic Publishing House
GENERALIZED STRONGLY SET-VALUED NONLINEAR
COMPLEMENTARITY PROBLEMS
NAN-JING HUANG and YEOL JE CHO
(Received 7 October 1997)
Abstract. In this paper, we introduce a new class of generalized strongly set-valued nonlinear complementarity problems and construct new iterative algorithms. We show the
existence of solutions for this kind of nonlinear complementarity problems and the convergence of iterative sequences generated by the algorithm. Our results extend some recent
results in this field.
Keywords and phrases. Complementarity problem, strongly monotone and Lipschitz continuous mappings, strongly monotone and H-Lipschitz continuous set-valued mappings,
algorithm.
1991 Mathematics Subject Classification. 90C33, 47H04.
1. Introduction. The complementarity theory, which was introduced by Lemke [11],
Cottle, and Dantzig [6] in the early 1960s and later developed by others, plays an important and fundamental role in the study of a wide class of problems arising in
mechanics, physics, control and optimization, economics and transportation equilibrium, contact problems in elasticity, fluid flow through porous media, and many other
branches of mathematical and engineering sciences [1, 2, 3, 4, 5, 7, 8, 9, 10, 13, 14, 15].
In particular, the set-valued quasi-(implicit)complementarity problems, considered
and studied by Chang and Huang [2, 3], are important among the generalizations
of the complementarity problems. In [14], Noor introduced and studied some new
classes of nonlinear complementarity problems for single-valued mappings in Rn and,
in [4], Chang and Huang introduced and studied some new nonlinear complementarity
problems for compact-valued fuzzy mappings and set-valued mappings which include
many kinds of complementarity problems, considered by Chang [1], Cottle et al. [7],
Isac [9], and Noor [13, 14], as special cases.
In this paper, we introduce and study a new class of generalized strongly set-valued
nonlinear complementarity problems and construct new iterative algorithms. We also
discuss the existence of solutions for this kind of nonlinear complementarity problems and the convergence of iterative sequences generated by the algorithm. Our
results improve and develop some results in [4, 13, 14].
2. Preliminaries. Let Rn be the Euclidean space endowed with norm · and inner
product (·, ·), respectively. In the sequel, we use the following notations:
n
2R = {A : A ⊂ Rn and A is nonempty},
n
n
CB(R ) = {A : A ⊂ R and A is nonempty, bounded, and closed},
(2.1)
(2.2)
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N.-J. HUANG AND Y. J. CHO
|x| = |x1 |, |x2 |, . . . , |xn |
for all x ∈ Rn .
(2.3)
n
Let F , G, Q : Rn → 2R be three set-valued mappings and f , g : Rn → Rn be two
single-valued mappings. Now, we consider the following problem.
Find u ∈ Rn , x ∈ F (u), y ∈ G(u), and z ∈ Q(y) such that
u ≥ 0,
f (x) + g(z) ≥ 0,
u, f (x) + g(x) = 0.
(2.4)
This problem is called the generalized strongly set-valued nonlinear complementarity problem.
If Q is the identity mapping on Rn , then problem (2.4) is equivalent to the following.
Find u ∈ Rn , x ∈ F (u) and y ∈ G(u) such that
u ≥ 0,
f (x) + g y ≥ 0,
u, f (x) + g y = 0.
(2.5)
This problem is called the generalized set-valued nonlinear complementarity problem.
If F is the identity mapping on Rn , then problem (2.5) is equivalent to the following.
Find u ∈ Rn and y ∈ G(u) such that
u ≥ 0,
f (u) + g y ≥ 0,
u, f (u) + g y = 0,
(2.6)
which was considered by Chang and Huang in [4].
If F = G is the identity mapping on Rn , then problem (2.5) is equivalent to the
following.
Find u ∈ Rn such that
u ≥ 0,
f (u) + g(u) ≥ 0,
u, f (u) + g(u) = 0,
(2.7)
which was considered by Noor [14].
If F = G is the identity mapping on Rn and f ≡ 0, then problem (2.5) is equivalent
to the following.
Find u ∈ Rn such that
u ≥ 0,
g(u) ≥ 0,
u, g(u) = 0,
(2.8)
which was considered by Karamardian [10], Fang [8], and Noor [13].
Obviously, problem (2.4) can be written as follows:
Find u ∈ Rn , x ∈ F (u), y ∈ G(u) and z ∈ Q(y) such that
u ≥ 0,
v = f (x) + g(z) ≥ 0,
(u, v) = 0.
We now consider the following equalities:
−1 u = 12 |w| + w ,
v = λρ
|w| − w ,
(2.9)
(2.10)
where λ, ρ > 0 are constants. Clearly, u ≥ 0 and v ≥ 0. From (2.5) and (2.9), it follows
that problem (2.9) is equivalent to the following.
Find w ∈ Rn , x ∈ F (1/2(|w| + w)), y ∈ G(1/2(|w| + w)), and z ∈ Q(y) such that
w = 12 |w| + w − 12 λρ f (x) + g(z) ,
(2.11)
where λ, ρ > 0 are constants.
GENERALIZED STRONGLY SET-VALUED NONLINEAR COMPLEMENTARITY
599
3. Algorithms. Based on the formulations in Section 2, we now construct the new
algorithms for the generalized strongly set-valued nonlinear complementarity problem (2.4).
Let F , G, Q : Rn → CB (Rn ) be three set-valued mappings and let f , g : Rn → Rn be
two single-valued mappings. For any wo ∈ Rn , let
x0 ∈ F
1
2
|w0 | + w0
,
y0 ∈ G
1
2
|w0 | + w0
,
z0 ∈ Q y0
(3.1)
and
1 1
w1 = 2 (1 − λ) |w0 | + w0 + 2 λ |w0 | + w0 − ρ f (x0 ) + g(z0 ) ,
(3.2)
where λ, ρ > 0 are constants.
Since x0 ∈ F (1/2(|w0 |+ w0 )), y0 ∈ G(1/2(|w0 | + w0 )) and z0 ∈ Q(y0 ), by Nadler
[12], there exist x1 ∈ F (1/2(|w1 |+w1 )), y1 ∈ G(1/2(|w1 |+w1 )) and z1 ∈ Q(y1 ) such
that
x0 − x1 ≤ (1 + 1)H F 12 |w0 | + w0 , F 12 |w1 | + w1
,
1 1
,
y0 − y1 ≤ (1 + 1)H G 2 |w0 | + w0 , G 2 |w1 | + w1
z1 − z0 ≤ (1 + 1)H Q y1 , Q y0 ,
(3.3)
(3.4)
(3.5)
where H(·, ·) denotes the Hausdorff metric on CB (Rn ).
Thus, by induction, we can obtain the following algorithm:
Algorithm 3.1. Let F , G, Q : Rn → CB (Rn ) be three set-valued mappings and let
f , g : Rn → Rn be two single-valued mappings. For any w0 ∈ Rn , we can construct
sequences {wn }, {xn }, {yn }, and {zn } in Rn as follows:
1
|wn | + wn ,
yn ∈ G 2 |wn | + wn ,
zn ∈ Q y n ,
1
xn − xn+1 ≤ 1 +
H F 12 |wn | + wn , F 12 |wn+1 | + wn+1
,
n+1
1
yn − yn+1 ≤ 1 +
H G 12 |wn | + wn , G 12 |wn+1 | + wn+1
,
n+1
1
H Q yn , Q yn+1 ,
zn − zn+1 ≤ 1 +
n+1
1 1
wn+1 = 2 (1 − λ) |wn | + wn + 2 λ |wn | + wn − ρ f (xn ) + g(zn )
xn ∈ F
1
2
(3.6)
for n = 0, 1, 2, . . . , where λ, ρ > 0 are constants.
From Algorithm 3.1, we can obtain the following algorithms.
Algorithm 3.2. Let F , G : Rn → CB (Rn ) be two set-valued mappings and let f , g :
Rn → Rn be two single-valued mappings. For any w0 ∈ Rn , we can construct sequences
600
N.-J. HUANG AND Y. J. CHO
{wn }, {xn }, and {yn } in Rn as follows:
1
1
yn ∈ G 2 |wn | + wn ,
xn ∈ F 2 |wn | + wn ,
1
xn − xn+1 ≤ 1 +
H F 12 |wn | + wn , F 12 |wn+1 | + wn+1
,
n+1
1
yn − yn+1 ≤ 1 +
H G 12 |wn | + wn , G 12 |wn+1 | + wn+1
,
n+1
wn+1 = 12 (1 − λ) |wn | + wn + 12 λ |wn | + wn − ρ f (xn ) + g(zn )
(3.7)
for n = 0, 1, 2, . . . , where λ, ρ > 0 are constants.
Algorithm 3.3 [4]. Let G : Rn → CB (Rn ) be a set-valued mapping and let f , g :
R → Rn be two single-valued mappings. For any w0 ∈ Rn , we can construct sequences
{wn } and {yn } in Rn as follows:
1
(3.8)
yn ∈ G 2 |wn | + wn ,
1
yn − yn+1 ≤ 1 +
H G 12 |wn | + wn , G 12 |wn+1 | + wn+1
,
(3.9)
n+1
wn+1 = 12 (1 − λ) |wn | + wn + 12 λ |wn | + wn − ρ f 12 |wn | + wn + g yn
(3.10)
n
for n = 0, 1, 2, . . . , where λ, ρ > 0 are constants.
Algorithm 3.4. [14] Let f , g : Rn → Rn be two single-valued mappings. For any
w0 ∈ Rn , we can construct sequences {wn } and {yn } in Rn as follows.
1
(3.11)
yn = 2 |wn | + wn ,
wn+1 = 12 (1 − λ) |wn | + wn
(3.12)
1
+ 2 λ |wn | + wn − ρ f 12 |wn | + wn + g yn
for n = 0, 1, 2, . . . , where λ, ρ > 0 are constants.
Algorithm 3.5. [15] Let g : Rn → Rn be a single-valued mapping. For any w0 ∈ Rn ,
we can construct sequences {wn } and {yn } in Rn as follows.
1
(3.13)
yn = 2 |wn | + wn ,
1 1
wn+1 = 2 (1 − λ) |wn | + wn + 2 λ |wn | + wn − ρ g yn
(3.14)
for n = 0, 1, 2, . . . , where λ, ρ > 0 are constants.
4. Existence and convergence. In this section, we show the existence of solutions
for the generalized strongly set-valued nonlinear complementarity problem (2.4) and
the convergence of the iterative sequences constructed by Algorithm 3.1. We first give
some definitions.
Definitions 4.1.
(1) A mapping f : Rn → Rn is said to be strongly monotone if there exists a constant
α > 0 such that
f (u) − f (v), u − v ≥ αu − v2
(4.1)
GENERALIZED STRONGLY SET-VALUED NONLINEAR COMPLEMENTARITY
601
for all u, v ∈ Rn .
(2) A mapping f : Rn → Rn is said to be Lipschitz continuous if there exists a constant
β > 0 such that
f (u) − f (v) ≤ βu − v
(4.2)
for all u, v ∈ Rn . The number β in (2) is called Lipschitz constant.
It is easy to see that α ≤ β.
Definitions 4.2.
(1) A set-valued mapping F : Rn → CB (Rn ) is said to be strongly monotone with
respect to the mapping f : Rn → Rn if there exists a constant α > 0 such that
f (x) − f y , u − v ≥ αu − v2
(4.3)
for all u, v ∈ Rn , x ∈ F (u) and y ∈ F (v).
(2) A set-valued mapping F : Rn → CB (Rn ) is said to be H-Lipschitz continuous if
there exists a constant β > 0 such that
H F (u), F (v) ≤ βu − v
(4.4)
for all u, v ∈ Rn . The number β in (2) is called the H-Lipschitz constant.
Now, we give our main theorems in this paper.
Theorem 4.1. Suppose that f , g : Rn → Rn are Lipschitz continuous with Lipschitz
constants δ and ξ, respectively, and F , G, Q : Rn → CB (Rn ) are H-Lipschitz continuous
with H-Lipschitz constants β, η, ν, respectively, and F is strongly monotone with respect
to f with strongly monotone constant α. If
0<ρ<
4(α − ξνη)
,
(δβ)2 − (ξνη)2
ρξνη < 2,
ξνη < min{α, δβ},
(4.5)
then there exist u, x, y, z ∈ Rn which solve problem (2.4). Furthermore, it follows that
1
|wn | + wn
2
→ u,
xn → x,
yn → y,
zn → z
as n → ∞,
(4.6)
where {wn }, {xn }, {yn }, and {zn } in Rn are the sequences generated by Algorithm 3.1.
Proof. By Algorithm 3.1, we have
wn+1 − wn = 12 (1 − λ) |wn | + wn + 12 λ |wn | + wn − ρ f xn + g zn
− 12 (1 − λ) |wn−1 | + wn−1 − 12 λ |wn−1 | + wn−1 − ρ f xn−1 + g zn−1 1 ≤ (1 − λ)wn − wn−1 + 2 λρ g zn − g zn−1 + λ 12 |wn | + wn − 12 |wn−1 | + wn−1 − 12 ρ f xn − f xn−1 .
(4.7)
602
N.-J. HUANG AND Y. J. CHO
Since F , G, and Q are H-Lipschitz continuous and f and g are Lipschitz continuous,
from (3.6), it follows that
f xn − f xn−1 ≤ δxn − xn−1 1
≤ δ 1+
H F 12 |wn | + wn , F 12 |wn−1 | + wn−1
n
(4.8)
|wn | + wn |wn−1 | + wn−1 1
≤ δ 1+
β
−
n
2
2
1
≤ δ 1+
βwn − wn−1 n
and
g zn − g zn−1 1
≤ ξ zn − zn−1 ≤ ξ 1 +
H Q yn , Q yn−1
n
1 ≤ ξν 1 +
yn − yn−1 n
1 2 1 H G 2 |wn | + wn , G 12 |wn−1 | + wn−1
≤ ξν 1 +
n
1 2 ηwn − wn−1 .
≤ ξν 1 +
n
(4.9)
Further, from the strong monotonicity of F with respect to f and (4.8), we have
1
2
|wn | + wn − 1 |wn−1 | + wn−1 − 1 ρ f xn − f xn−1 2
2
2
2 2
1
1
≤ 1 − αρ + ρ 2 δ2 1 +
β2 wn − wn−1 .
4
n
(4.10)
Thus, it follows, from (4.7), (4.8), (4.9), and (4.10), that
wn+1 − wn 1
1
1 2 2 2
1 2 wn − wn−1 ≤ 1 − λ + λρξη 1 +
+ λ 1 − αρ + ρ δ β 1 +
2
n
4
n
= θn wn − wn−1 ,
(4.11)
where
1 2
1
+λ
θn = 1 − λ + λρξνη 1 +
2
n
1
1 2
1 − αρ + ρ 2 δ2 β2 1 +
.
4
n
(4.12)
Letting
1
θ = 1 − λ + λρξνη + λ
2
1
1 − αρ + ρ 2 δ2 β2 ,
4
(4.13)
θn → θ as n → ∞. In view of (4.5), we know that 0 < θ < 1 and so θn < 1 for n
sufficiently large. It follows from (4.11) that {wn } is a Cauchy sequence in Rn and so
GENERALIZED STRONGLY SET-VALUED NONLINEAR COMPLEMENTARITY
603
we can suppose that wn → w as n → ∞. (4.8) and (4.9) imply that {xn }, {yn }, and {zn }
are also Cauchy sequences in Rn . Let xn → x, yn → y, and zn → z as n → ∞. Letting
u = 12 (|w| + w), we get
1
|wn | + wn
2
→ u,
xn → x,
yn → y,
zn → z
as n → ∞.
(4.14)
Now, we prove that x ∈ F (u), y ∈ G(u), and z ∈ Q(y). In fact, we have
d(x, F (u)) = inf{x − z : z ∈ F (u)}
≤ x − xn + d xn , F (u)
≤ x − xn + H F 12 wn + wn , F (u)
1
≤ x − xn + β 2 wn + wn − u,
(4.15)
and hence d(x, F (u)) = 0. This implies that x ∈ F (u). Similarly, we have y ∈ G(u)
and z ∈ Q(y). This completes the proof.
From Theorem 4.1, we get the following theorem.
Theorem 4.2. Suppose that f , g : Rn → Rn are Lipschitz continuous with Lipschitz
constants δ and ξ, respectively, and F , G : Rn → CB (Rn ) are H-Lipschitz continuous with
H-Lipschitz constants β and η, respectively, and F is strongly monotone with respect to
f with strongly monotone constant α. If
0<ρ<
4(α − ξη)
,
(δβ)2 − (ξη)2
ρξη < 2,
ξη < min{α, δβ},
(4.16)
then there exist u, x, y ∈ Rn which solve problem (2.5). Furthermore, it follows that
1 wn + wn
2
→ u,
xn → x,
yn → y
as n → ∞,
(4.17)
where {wn }, {xn }, and {yn } in Rn are the sequences generated by Algorithm 3.2.
Theorem 4.3 [4]. Let f , g, and G be the same as in Theorem 4.2. Further, suppose
that f is strongly monotone with strongly monotone constant α. If
0<ρ<
4(α − ξη)
,
δ2 − (ξη)2
ρξη < 2,
ξη < α,
(4.18)
then there exist u, y ∈ Rn which solve problem (2.6), and
1
(|wn | + wn )
2
→ u,
yn → y,
as n → ∞,
(4.19)
where {wn } and {yn } are two sequences generated by Algorithm 3.3.
Theorem 4.4 [14]. Let f and g be the same as in Theorem 4.3. If
0<ρ<
4(α − ξ)
,
δ2 − ξ 2
ρξ < 2,
ξ < α,
then there exists u ∈ Rn which is a solution of problem (2.7), and
n → ∞, where {wn } is the sequence generated by Algorithm 3.4.
(4.20)
1
(|wn | + wn )
2
→ u as
604
N.-J. HUANG AND Y. J. CHO
Theorem 4.5 [15]. Suppose that g : Rn → Rn is strongly monotone and Lipschitz
continuous with strongly monotone constant α and Lipschitz constant δ. If 0 < ρ <
4α/δ2 , then there exists u ∈ Rn which is a solution of problem (2.8), and 12 (|wn |+wn ) →
u as n → ∞, where {wn } is the sequence generated by Algorithm 3.5.
Acknowledgement. The second author was supported by the academic research
fund of Ministry of Education, Republic of Korea, 1997, Project No. BSRI-97-1405.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
S. S. Chang, Variational Inequality and Compltementarity Problem Theory with Applications, Shanghai Scientific and Tech. Literature Publishing House, Shanghai, 1991.
S. S. Chang and N. J. Huang, Generalized multivalued implicit complementarity problems in Hilbert spaces, Math. Japon. 36 (1991), no. 6, 1093–1100. MR 92m:47139.
Zbl 748.49006.
, Generalized strongly nonlinear quasi-complementarity problems in Hilbert spaces,
J. Math. Anal. Appl. 158 (1991), no. 1, 194–202. MR 92d:90104. Zbl 739.90067.
, Generalized complementarity problems for fuzzy mappings, Fuzzy Sets and Systems 55 (1993), no. 2, 227–234. MR 94e:65067. Zbl 790.90076.
R. W. Cottle, Complementarity and variational problems, Symposia Mathematica, 19 (London), Academic Press, 1976, pp. 177–208. MR 58 20486. Zbl 349.90083.
R. W. Cottle and G. B. Dantzig, Complementary pivot theory of mathematical programming, Linear Algebra Appl. 1 (1968), no. 1, 103–125. MR 37#2515. Zbl 155.28403.
R. W. Cottle, J. S. Pang, and R. E. Stone, The Linear Complementarity Problem, Computer
Science and Scientific Computing, Academic Press, Inc., Boston, London, 1992.
MR 93f:90001. Zbl 757.90078.
S. C. Fang, An iterative method for generalized complementarity problems, IEEE Trans.
Automat. Control 25 (1980), no. 6, 1225–1227. MR 82i:90116. Zbl 483.49027.
G. Isac, Complementarity Problems, Lecture Notes in Mathematics, vol. 1528, SpringerVerlag, Berlin, 1992. MR 94h:49002. Zbl 795.90072.
S. Karamardian, Generalized complementarity problem, J. Optim. Theory Appl. 8 (1971),
161–168. MR 47 10073. Zbl 218.90052.
C. E. Lemke, Bimatrix equilibrium points and mathematical programming, Management
Sci. 11 (1964/1965), 681–689. MR 32#7243. Zbl 139.13103.
S. B. Nadler, Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475–488.
MR 40#8035. Zbl 187.45002.
M. A. Noor, Generalized nonlinear complementarity problem, J. Natur. Sci. Math. 26 (1986),
no. 1, 9–19. MR 87k:90265. Zbl 595.90088.
, Fixed point approach for complementarity problems, J. Math. Anal. Appl. 133
(1988), no. 2, 437–448. MR 89g:90223. Zbl 649.65037.
M. A. Noor and S. Zarae, An iterative scheme for complementarity problems, Engrg. Anal.
J. 3 (1986), 240–243.
Huang: Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064,
China
Cho: Department of Mathematics, Gyeongsang National University, Chinju 660-701,
Korea